Method and apparatus for predicting transient response of a closed loop apparatus

ABSTRACT

A method for predicting transient response of a closed loop apparatus includes the steps of: (a) providing a first reference tool that relates load-free impedance response with a first design gain-phase variable; (b) providing a second reference tool that relates load-free impedance response with a second design gain-phase variable; (c) determining a combined impedance response as a function of frequency; (d) employing at least one of the first and second reference tool to establish a first design value for one of the phase variable and the design load impedance at a characteristic frequency that occurs at a peak value of the combined impedance response; (e) employing at least one of the first and second reference tool to establish a second design value for the other parameter of the phase variable and the design load impedance at the characteristic frequency; (f) establishing a transient multiplier as a function of frequency associated with the output voltage with the design load impedance for selected values of phase margin; (g) creating a third reference tool relating the transient multiplier with phase margin; (h) employing the third reference tool to establish a third design value for the transient multiplier associated with the characteristic frequency and at least one of the first design and second design value; and (i) mathematically combining at least two of the first, second and third design value with a design step current to establish transient excursion of the output voltage in response to applying the design step current.

This is a continuation-in-part of application U.S. Ser. No. 09/609,044,filed Jun. 30, 2000 now U.S. Pat. No. 6,760,633.

BACKGROUND OF THE INVENTION

The present invention is directed to a method and apparatus forpredicting transient response characteristics of power supplies or otherclosed loop systems under arbitrary load conditions. The presentinvention is particularly directed to a method for predicting transientresponse characteristics for direct current, DC-to-DC, power supplies.

When designing certain systems, such as power supply, or power convertercircuits, one must take into account the potential user's loadcharacteristics. This consideration is especially important in thedesign of DC-DC converters because such converters are generallyconfigured as a closed loop system that monitors its output, providesfeedback indicating its output, and employs the feedback to adjust tomaintain a constant DC output. In any feedback system, it is ofsignificant importance that the feedback loop be stable. A simpleexample of an unstable feedback loop is the loud tone produced in thepresence of audio feedback when a microphone is placed too close to aspeaker producing signals originating at the microphone.

Today's electronic devices are more and more designed to be faster,smaller, and more reliable. This trend for product requirements isespecially evident in portable electronic devices such as cellulartelephones, electronic games, and portable computers. Some practicaldesign consequences of this trend are that output voltages for DC-DCconverters are getting lower and the stability of output of DC-DCconverters is more difficult to attain for certain loads orapplications.

The fact that a user's load characteristics figure so intimately instability of DC-DC converter circuits, and the ever more stringentrequirements for greater stability at lower voltages for modernelectronic circuits have made present ways of predicting stability of aparticular DC-DC converter circuit for a particular applicationuneconomical and not particularly reliable or accurate.

Nyquist developed criteria to assess the stability of a control loop(“regeneration Theory”, H. Nyquist, Bell System Technical Journal,January 1932). Bode (“Relations Between Attenuation and Phase inFeedback Amplifier Design”, Bell System Technical Journal, July 1940)expressed these criteria in terms of the phase (φ) and gain of atransfer function. According to this analysis, if gain (dB) and phasechange (Δφ) of the loop gain are zero at the same frequency in acircuit, the circuit will be unstable.

As a practical engineering measure, one must design a circuit having≧45° phase margin to reliably have a stable circuit. Phase margin is thevalue of phase when gain as a function of frequency crosses through zerofrom positive to negative. Thus, when gain is 0 dB, and gain is passingfrom positive to negative, phase must be ≧45° in order for the circuitunder consideration to be stable with adequate margin.

Another measure of stability is to require that gain margin be ≧−7 to−10 dB. That is, when phase as a function of frequency crosses throughzero, gain must be at least 7-10 dB in order that the circuit underconsideration will be a stable circuit.

Presently, manufacturers of power supplies, and especially of DC-DCconverters, use simulations, or laboratory measurements, or closed formanalytical expressions, or all three of those methods for determiningwhether a particular circuit is stable with a particular load.Simulations are expensive in that they occupy large amounts of computercapacity and time. Closed form analytical expressions rely onsimplifying assumptions that introduce significant errors. Laboratorymeasurements are an expensive approach to answering questions about aparticular circuit-load stability in terms of human time and computerassets involved. Further, neither simulations, closed form analyticalexpressions nor laboratory experimentation are particularly accurate inpredicting stability of converter apparatuses under various loadconditions.

One result of ongoing efforts to predict stability with arbitrary loadsis that manufacturers of power converters must essentially custom-tailortheir products to user's loads on a case-by-case basis. Such a “jobshop” approach to production precludes one's taking advantage of theeconomies of scale which could be enjoyed if a manufacturer couldpredict which loads were amenable to stable use with particularconverters. That is, if manufacturers could predict stability for aparticular converter circuit for a particular load without having tophysically evaluate the converter circuit with the particular load, thenthe inefficiencies of customizing converter circuits for each discreteload criterion may be avoided.

A product designer is concerned with stability of the circuits that areincorporated in the products, but must also be concerned with thetransient response characteristics of the circuits. That is, there mustbe consideration of the transient voltage characteristics and thesettling time of the design. Settling time for a circuit or system isthe time that lapses from a perturbation of the system until theparameter being measured (e.g. output voltage in the case of a typicalpower supply) is within a desired percentage of a desired value.Specifically, by way of example, settling time for a power supply may bethe time for a power supply's output voltage to return to within 1-2% ofa desired design output voltage for the power supply after the powersupply is switched on. The transient voltage is the amplitude excursionof the output voltage during the settling time interval.

If manufacturers could also predict transient response characteristicsfor a particular converter circuit for a particular load without havingto actually physically evaluate the converter circuit with theparticular load, they would enjoy an added advantage in predictabilityof critical operational characteristics of their products.

Moreover, one may use the present invention to ascertain designcharacteristics necessary for products to exhibit certain operationalcharacteristics. For example, one may determine that a product desirablyshould operate with a particular transient peak voltage and settlingtime for a particular current step-change. Using the present invention,a designer may determine appropriate design parameters for the desiredproduct operating characteristics.

Manufacturers enjoying such an advantage in predictability ofoperational characteristics of their products vis-à-vis loads mayproduce converter apparatuses for “off-the-shelf” availability tocustomers with evaluation tools enabling customers to select which ofthe converters will accommodate the particular loads they are designing.

There is a need for a method for predicting transient responsecharacteristics of power converters under arbitrary load conditions.This need is particularly acute in predicting transient responsecharacteristics of DC-DC power converter circuits.

It would be particularly useful if transient response characteristics ofpower supply apparatuses could be predicted without having to test thepower supply apparatus under the particular load condition for which atransient response determination is desired.

The method of the present invention allows evaluation of the transientresponse of a power supply apparatus for various load conditions withouthaving to recharacterize the apparatus for each given load.

SUMMARY OF THE INVENTION

A method for predicting at least one transient response characteristicof a closed loop apparatus having an open loop impedance, a design loadimpedance, an output voltage and at least one inherent internal gainincludes the steps of: (a) providing a first reference tool that relatesimpedance response of the apparatus, independent of the design loadimpedance, with a first variable of a gain variable and a phasevariable; (b) providing a second reference tool that relates impedanceresponse of the apparatus, independent of the design load impedance,with a second variable of a gain variable and a phase variable otherthan the first variable; (c) determining a combined impedance responsefor the apparatus as a function of frequency; the combined impedanceresponse involving the open loop impedance, the design load impedanceand the at least one inherent internal gain; (d) employing at least oneof the first and second reference tool to establish a first design valuefor one parameter of the phase variable and the design load impedance ata characteristic frequency; the characteristic frequency occurringsubstantially at a peak value of the combined impedance response; (e)employing at least one of the first and second reference tool toestablish a second design value for the other parameter of the phasevariable and the design load impedance than the one parameter at thecharacteristic frequency, (f) vectorally establishing a transientmultiplier factor that is associated with the output voltage with thedesign load impedance as a function of frequency for selected values ofphase margin; (g) creating a third reference tool relating the transientmultiplier factor with the selected values of phase margin as a functionof frequency; (h) employing the third reference tool to establish athird design value for the transient multiplier factor associated withthe characteristic frequency and at least one of the first design valueand the second design value; and (i) mathematically combining at leasttwo of the first design value, the second design value and the thirddesign value with a design step current according to a predeterminedrelation to establish transient voltage excursion of the output voltagesubstantially at the characteristic frequency in response to applyingthe design step current.

The apparatus includes a first reference tool relating a first impedancescaling value with a first design variable; a second reference toolrelating a second impedance scaling value with a second design variable;and a third reference tool relating the design load impedance with athird design variable.

The stability of a controlled apparatus, that is an apparatus withregeneration or feedback, such as a regulated power supply, powerconverter, amplifier or other closed loop apparatus, is an important, ifnot critical, consideration in any application of that apparatus.Measures of the stability or potential stability of a controlledapparatus include the phase margin and the gain margin. Preferably, boththe phase margin and the gain margin of an apparatus are considered inevaluating the stability of the apparatus. Such margin measures are anindication of how close the control system or the loop response of thatapparatus is to instability. The loop response itself is a function ofthe load placed on the output of such an apparatus.

The conventional approach to evaluate or determine the margins of suchan apparatus has been to generate a Bode plot of the loop response for aspecific load condition. By inspection of such a Bode plot one maydetermine the value of the margin of the apparatus being evaluated forthat specific load condition. In the case where the load is to bedesigned appropriately to maintain the apparatus in a stable conditionduring operation, the conventional approach has resulted in a timeconsuming process of iterations of load adjustments, Bode plotgeneration for each adjustment, inspection and readjustment. By suchiterative employment of the conventional approach, one may step-wiseascertain a load that permits stable operation of an apparatus.

The preferred embodiment of the present invention produces a responseplot of a closed loop apparatus that is not dependent on the loadcharacteristics with which the apparatus is to be employed for the basicplot generation. As a result, the same plot can be used to determine theoperating margin of the apparatus characterized by the plot for anyvariation of the load with which the apparatus is to be employed. Such aload-independent evaluation method can significantly reduce the effortof characterizing the response of a power supply apparatus for a givenload.

Another characteristic of interest to product designers and other usersof closed loop devices, such as power converters, is transient responsecharacteristics of the devices. Transient response characteristicsinclude two aspects: (1) transient voltage (e.g., how the closed loopsystem responds to imposition of a perturbation, such as a step-changein current); and (2) settling time (e.g., how long the system takes tosettle to within a predetermined amount of a quiescent value afterimposition of the perturbation). The teachings of the present inventionare broadly applicable to any closed loop system, including for examplefluid systems; air conditioning systems, television sets, food mixersand other home appliances; engine governors and controllers; powergeneration systems for a city and other systems. The teachings of thepresent invention are particularly applicable to power convertercircuits and apparatuses.

Features of the present invention will be apparent from the followingspecification and claims when considered in connection with theaccompanying drawings, in which like elements are labeled using likereference numerals in the various figures, illustrating the preferredembodiment of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an electrical schematic diagram of a power converter apparatusconnected with a resistive load.

FIG. 2 is an exemplary Nyquist Plot of real and imaginary parts of loopgain in a closed loop apparatus.

FIG. 3 is an exemplary Bode Plot of magnitude and phase of loop gainplotted as a function of frequency for a closed loop apparatus.

FIG. 4 is an exemplary plot of contours of constant phase margin plottedon axes representing complex load impedance of a closed loop apparatusin terms of capacitance and equivalent series resistance.

FIG. 5 is an exemplary plot generated according to the preferredembodiment of the method of the present invention.

FIG. 6 is an electrical schematic diagram of a power converter apparatusthat is a generalized equivalent to the circuit illustrated in FIG. 1.

FIG. 7 is an electrical schematic diagram of a power converter of thesort illustrated in FIG. 6, configured for evaluating transient outputvoltage characteristics.

FIG. 8 is a graphic plot as a function of frequency of variousimpedances in the closed loop system of FIG. 7.

FIG. 9 is an electrical schematic diagram of a resonant circuit.

FIG. 10 is a graphic plot of signal amplitude as a function of time,illustrating various parameters associated with describing a signalresponse.

FIG. 11 is a graphic plot of multiplier: Y, as a function of frequencyfor various values of phase margin.

FIG. 12 is a graphic plot of settling factor at the characteristicfrequency of a system as a function of phase margin.

FIGS. 13(A) and (B) illustrate application of one aspect of the methodof the present invention.

FIGS. 14(A) and (B) illustrate application of a second aspect of themethod of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 is an electrical schematic diagram of a power converter apparatusconnected with a resistive load. Thottuvelil and Verghese characterizeda power converter using the power converter apparatus of FIG. 1 as aThevenin voltage source model in their paper setting forth asmall-signal stability analysis of paralleled DC-DC converter systems.(See, V. Joseph Thottuvelil and George C. Verghese; “Analysis andControl Design of Paralleled DC/DC Converters with Current Sharing”;IEEE Transactions on Power Electronics, Vol. 13, No. 4; July 1998.).

In principle, the load indicated in FIG. 1 could as well be a complexload. A resistive load is used to illustrate the method of the presentinvention in connection with FIG. 1 to simplify the explanation. In FIG.1, a power converter apparatus 10 includes a Thevenin voltage source 12providing a voltage V_(TH) and connected in series with an open-loopoutput impedance 15. Open-loop output impedance 15 has a value ofZ_(OL). Converter apparatus 10 has two output loci, or terminals 26, 28and a sense locus, or terminal 30. Output terminals 26, 28 are connectedwith an output circuit 40. Output circuit 40 includes a resistive load38 connected across output terminals 26, 28. Resistive load 38 has avalue R_(L).

A reference voltage V_(REF) is applied to a positive input node 14 of adifference generator 16. Difference generator 16 also receives, at anegative input node 22, a feedback signal multiplied by a gain stage 36having a gain B, via a sense line 20 from output side 18 of open-loopoutput impedance 15, via output terminal 26 and via sense terminal 30.The difference between reference voltage V_(REF) at positive input node14 and the feedback signal at negative input node 22 is provided as acontrol signal by difference generator 16 from an output node 17 via aline 24 subject to a gain A, represented by a box 34, to controlThevenin voltage source 12. The control signal provided from output node17 of difference generator 16 keeps Thevenin voltage source 12generating a signal having a voltage value V_(TH). Thevenin voltagesource 12 provides voltage V_(TH) to open-loop impedance 15 from anoutput node 13 via a line 32. Gain A, represented schematically at box34, is the gain from output node 17 of difference generator 16 to outputnode 13 of Thevenin voltage source 12. Gain A and Gain B are inherentinternal gains of power converter apparatus 10. Gain A does not includethe effect of open-loop output impedance 15, or any external loadimpedances. Gain B, represented schematically at box 36, is the gainfrom sensed voltage at output terminal 26 to negative input node 22 ofdifference generator 16.

In a power supply apparatus such as power converter apparatus 10illustrated in FIG. 1, an important design objective is to limitvariations in V_(L) relative to V_(REF). The variation of V_(L) withrespect to V_(REF) is given by the equation: $\begin{matrix}{\frac{V_{o}}{V_{R}} = \frac{\gamma\quad A}{1 + {\gamma\quad{AB}}}} & \lbrack 1\rbrack\end{matrix}$

-   -   where γ is given by: $\begin{matrix}        {\gamma = \frac{Z_{L}}{Z_{L} + Z_{OL}}} & \lbrack 2\rbrack        \end{matrix}$    -   where Z_(OL) is open loop impedance;        -   Z_(L) is load impedance (illustrated as resistive load R_(L)            in FIG. 1).

The second term of the denominator of Equation [1], in conventionalcontrol theory, is recognized as the loop gain. That is, the loop gain(LG) is given by the equation: $\begin{matrix}{{LG} = {{AB}\frac{Z_{L}}{Z_{L} + Z_{OL}}}} & \lbrack 3\rbrack\end{matrix}$

It is important to recognize that the loop gain is a function of loadimpedance Z_(L). As a consequence, any time load impedance Z_(L) ischanged, loop gain is changed.

FIG. 2 is an exemplary Nyquist Plot of real and imaginary parts of loopgain in a closed loop apparatus. In general, loop gain is a complexfunction; that is, a function containing real and imaginary components.In 1932, Nyquist (Bell System Technical Journal, January 1932)introduced a graphical means of observing or characterizing thestability of control loops. FIG. 2 illustrates such a “Nyquist Plot”.Nyquist found that as the real (Re) and imaginary (Im) parts of the loopgain of a system were plotted as a function of frequency (f) from zeroto infinity, if the resulting curve did not contain “−1” (as representedby curve I in FIG. 2), then the system would be stable. If the resultingcurve did contain “−1” (as represented by curve II in FIG. 2), thesystem would be unstable. This is consistent with Equation [1]; if theterm γAB=−1, the expression goes to ∞. Such a condition indicatesinstability. A resulting curve of the sort represented by curve III inFIG. 2 was said to be conditionally stable.

The “gain margin” of a system is defined as the distance along the real(Re) axis between “−1” and the loop gain curve intersection with thereal (Re) axis. The “phase margin” of a system is defined as the anglebetween the real (Re) axis and the intersection of the loop gain curvewith unit gain circle 50. That is, gain margin is measured at a locus atwhich phase margin is 0°, and phase margin is measured at a locus atwhich gain margin is 1. Thus, for example in FIG. 2, curve I will have again margin of Δ, and will have a phase margin of θ.

As a system response approaches −1 on the real axis Re, oscillationswithin the system increase; that is, system instability increases. Suchoscillations, or perturbations, may be caused by such influences asthermal noise, load changes (occasioned by, for example, thermaleffects, on/off switching, and changes in switching states), switchingnoise or other influences.

System designers seeking to design systems that operate robustly andreliably generally observe system design rules to provide for stablesystem operation. These design rules are intended to allow for suchfactors as manufacturing variances, degradation of components with time,thermal variance, and other factors. Generally accepted system designrules for power supply apparatuses require phase margin ≧45°, and gainmargin ≧−7 to −10 dB.

FIG. 3 is an exemplary Bode Plot of magnitude and phase of loop gainplotted as a function of frequency for a closed loop apparatus. In 1940,Bode (Bell System Technical Journal, July 1940), introduced anothergraphical method that is used to evaluate system stability. Bode'smethod is also covered in U.S. Pat. No. 2,123,178.

In FIG. 3 a “Bode plot” is illustrated in which magnitude 66 (in dB) andphase 68 (in degrees) of loop gain of an exemplary system are plotted asa function of frequency (in kHz). In the Bode Plot of FIG. 3, the gainmargin ΔdB is defined as the gain where phase is zero degrees (i.e., atpoint 61). Thus, gain margin ΔdB is indicated at point 60 on themagnitude plot 66 of FIG. 3, at frequency f₁. The phase margin Δφ isdefined as the phase where the gain is unity or 0 dB (i.e., at point63). Thus, phase margin Δφ is indicated at point 62 on the phase plot 68of FIG. 3, at frequency f₂.

In practice, the difficulty with Nyquist Plots (FIG. 2) and Bode Plots(FIG. 3) is that loop gain (LG) changes with changes in load (Z_(L)).Because loop gain (LG) changes, the plots (Nyquist Plots and Bode Plots)also change with changes in load (Z_(L)). So, as load (Z_(L)) isadjusted, a new Nyquist Plot or Bode Plot must be generated in order toevaluate the system with the new load (Z_(L)). This is a tedious andtime consuming process.

FIG. 4 is an exemplary plot of contours of constant phase margin plottedon axes representing complex load impedance of a closed loop apparatusin terms of capacitance and equivalent series resistance.

In 1995, Rozman and Fellhoelter, (“Circuit Considerations for Fast,Sensitive, Low-Voltage Loads in a Distributed Power System”, APEC 1995Conference Proceedings, pg. 34) recognized the difficulty in the tediousapplication of Nyquist Plots and Bode Plots in evaluating stability ofclosed loop apparatuses, such as power supply apparatuses, for differentloads. Rozman and Fellhoelter introduced another graphical method forevaluating system stability.

Rozman and Fellhoelter assumed that the significant part of loadimpedance Z_(L) can be described as a capacitor having a capacitance Cand an equivalent series resistance ESR. In FIG. 4, a representativeRozman and Fellhoelter Plot is a contour plot wherein the two axes are ahorizontal axis 70 relating to load capacitance C and a vertical axis 71relating to equivalent series resistance ESR. Contour lines representloci of constant phase margin Δφ.

Thus, in FIG. 4, loci of a constant phase margin of 75° are representedby a curve 72. Loci of a constant phase margin of 60° are represented bya curve 74. Loci of a constant phase margin of 45° are represented by acurve 76. Loci of a constant phase margin of 30° are represented by acurve 78.

Rozman and Fellhoelter further contemplated a similar two-dimensionalplot on axes relating to capacitance C and equivalent series resistanceESR establishing contour lines representing loci of constant gainmargin. A plot of loci of constant gain margin according to theteachings of Rozman and Fellhoelter is considered within theunderstanding of one skilled in the art relevant to the subject matterof the present application. In the interest of avoiding prolixity, sucha representative plot is not included in this application.

The Rozman and Fellhoelter Plots (loci of constant phase margin and lociof constant gain margin) are an efficient method for quickly determiningthe stability of a system in terms of gain margin or phase margin, giventhe capacitance C and equivalent series resistance ESR of a load. Thedisadvantage of this approach is that its requisite configuration of theload is too inflexible. That is, the Rozman and Fellhoelter Plot isrestricted to loads that can be fairly characterized by a capacitor Cand an associated equivalent series resistance ESR Real world loads areoften too complex to be accurately approximated, or represented by acapacitor C and an associated equivalent series resistance ESR. Forexample, some loads may include different types or values of capacitorsin parallel, some loads may exhibit different responses at differenttemperatures, and some loads may exhibit different responses atdifferent frequencies. Such differences in load configuration involvecomplicating factors precluding accurate approximation of load responseby a simple capacitance C and equivalent series resistance ESR.

The present inventor has concluded that stability analysis focuses onparticular operating conditions. For example, the present inventorconcluded that phase margin analysis does not need to be evaluated for arange of values of gain. Instead, phase margin analysis need only beconcerned with loop gains equal to unity. Similarly, gain marginanalysis need only be concerned with operations at 0° phase.

As is evident with a Bode Plot (FIG. 3), if one looks solely at theunity gain point (FIG. 3, point 63), the resulting phase is defined asthe phase margin Δφ (FIG. 3, point 62). For gain margin analysis, one isonly concerned with the point of zero phase margin (FIG. 3, point 61).In FIG. 3, the resulting negative gain is the gain margin ΔdB, asindicated at point 60.

In view of such a recognition that the only relevant loci of concern inevaluating phase margin and gain margin of a system are, respectively,loci of unity gain and zero phase, Equation [3] may be employed todefine an important new term: $\begin{matrix}{{LG} = {{{AB}\frac{Z_{L}}{Z_{L} + Z_{OL}}} = {G\quad{\mathbb{e}}^{j\theta}}}} & \lbrack 4\rbrack\end{matrix}$

-   -   where G is the magnitude of the gain and e^(jθ) accounts for        phase margin (θ). Equation [4] can be rewritten as:        $\begin{matrix}        {Z_{L} = \frac{Z_{OL}}{\left\lbrack {{\frac{AB}{G}{\mathbb{e}}^{- {j\theta}}} - 1} \right\rbrack}} & \lbrack 5\rbrack        \end{matrix}$

It is important to note that Equation [5] has all load information onone side of the equal sign, and all other information on the other sideof the equal sign.

The present invention, as it is employed for evaluating phase margin ofa system, plots the right side of Equation [5] with gain magnitude G=1(i.e., at a unity gain) in terms of magnitude and phase as a function offrequency. Multiple curves are generated on the same graph representingsteps in values of phase margin. The steps in phase margin may be equalor not equal. The present inventor has determined that equal steps inphase margin plotting are preferred in order to facilitate interpolativeemployment of the plots for evaluation of a system vis-à-vis aparticular load. FIG. 5 is one example of a reference tool based upon avectoral determination of the right hand side of Equation [5] todetermine magnitude and phase response of a system as a function offrequency, with gain magnitude G=1.

FIG. 5 is an exemplary plot generated according to the preferredembodiment of magnitude and phase characteristics of the right side ofEquation [5] for a particular system, such as a power supply apparatus,is presented. The response curves developed and plotted in FIG. 5 areindependent of load characteristics of the system. Thus, FIG. 5 is aphase margin evaluative tool.

A gain margin graphic evaluative tool may be similarly produced. In sucha case, the present invention plots the right side of Equation [5] interms of magnitude and phase as a function of frequency, with phasemargin held at a value of 0°. Multiple curves, on the same graph aregenerated representing steps in values of gain margin. The steps in gainmargin may be equal or not equal. The present inventor has determinedthat equal steps in gain margin plotting are preferred in order tofacilitate interpolative employment of the plots for evaluation of asystem vis-à-vis a particular load.

For ease of reference, the right hand side of Equation [5] will bereferred to hereinafter as a Margin Function. When the magnitude andphase of the right hand side of Equation [5] is determined for a system,with gain magnitude G=1, the Margin Function will be referred to as aPhase Margin Function. When the magnitude and phase of the right handside of Equation [5] is determined, with phase=0°, the Margin Functionwill be referred to as a Gain Margin Function.

In FIG. 5, Phase Margin Functions are plotted for an array of phasemargin values. A Phase Margin Function indicating magnitude as afunction of frequency for a phase margin of 90° is plotted as responsecurve 82 a; a Phase Margin Function indicating phase for a phase marginof 90° is plotted as response curve 82 b. A Phase Margin Functionindicating magnitude as a function of frequency for a phase margin of75° is plotted as response curve 84 a; a Phase Margin Functionindicating phase for a phase margin of 75° is plotted as response curve84 b. A Phase Margin Function indicating magnitude as a function offrequency for a phase margin of 60° is plotted as response curve 86 a; aPhase Margin Function indicating phase for a phase margin of 60° isplotted as response curve 86 b. A Phase Margin Function indicatingmagnitude as a function of frequency for a phase margin of 45° isplotted as response curve 88 a; a Phase Margin Function indicating phasefor a phase margin of 450 is plotted as response curve 88 b. A PhaseMargin Function indicating magnitude as a function of frequency for aphase margin of 30° is plotted as response curve 90 a; a Phase MarginFunction indicating phase for a phase margin of 30° is plotted asresponse curve 90 b. A Phase Margin Function indicating magnitude as afunction of frequency for a phase margin of 15° is plotted as responsecurve 92 a; a Phase Margin Function indicating phase for a phase marginof 15° is plotted as response curve 92 b.

Other reference tools maybe produced representing similar vectoralexercising of the right side of Equation [5] to facilitate utility ofthe information thereby gleaned. Examples of such alternate embodimentsof reference tools include multidimensional vectoral tables and otherarrays of data resulting from such a vectoral exercising of the rightside of Equation [5]. Such reference tools, including multidimensionalvectoral tables, may be maintained on-line within a computer device forfacilitating mathematical or other algorithmic manipulation andevaluation of the data contained within the reference tool.

As an example of a use of the reference tool illustrated in FIG. 5, onemay evaluate phase margin response of a system with a particular load byplotting the load magnitude response curve 100 and load phase responsecurve 102 for the particular test load on the phase margin evaluativetool illustrated in FIG. 5. Load magnitude response curve 100 intersectsPhase Margin Function curve 82 a (indicating Phase Margin Functionmagnitude when gain magnitude G=1; Equation [5]) at a locus 104. Locus104 occurs at a sample frequency substantially equal to a value of 2.9kHz. Applying that sample frequency value (2.9 kHz) to Phase MarginFunction curves 82 b, &4 b, 86 b, 88 b, 90 b, 92 b (indicating PhaseMargin Function phase when gain magnitude G=1; Equation [5]1) one mayobserve that the sample frequency (2.9 kHz) intersects load phaseresponse curve 102 at a locus 106. Locus 106 is situated between PhaseMargin Function curves 84 b, 86 b. Interpolating between Phase MarginFunction curves 84 b, 86 b one may conclude that the system has a phasemargin slightly less than 75°, approximately 73°, when employed with theparticular load represented by load magnitude response curve 100 andload phase response curve 102.

The reference tool embodied in FIG. 5 enables quick evaluation of systemresponse with a given load to determine whether the system will meetdesign criteria (e.g., phase margin ≧45°; gain margin ≧−7 to −−10 dB)when employed with the given load. No reconstruction of the evaluativetool (FIG. 5) is necessary to conduct an evaluation of the system withanother load. Similar flexibility and ease of use is afforded by gainmargin evaluative tools created using the method of the presentinvention.

There are at least two key advantages provided by the method andapparatus of the present invention over prior art conventionalapproaches previously described in connection with FIGS. 2-4. First, theevaluation tool does not need to be reconstructed or changed in order toevaluate iterations or changes in the load to be employed with thesystem. The evaluation tool may be employed to evaluate any load thatcan be characterized in terms of a frequency dependent magnitude andphase. Second, the evaluation tool gives a “fingerprint” that ischaracteristic of that particular power supply, amplifier or otherclosed loop apparatus for any load condition. This advantage isparticularly useful, for example, in comparing power supply systems inorder to determine which system may be better suited for handling aparticular load In such a situation, plots of the two (or more) powersupplies can be quickly and straightforwardly compared to determinewhich power supply would be more stable for a given load condition.

The inventor has discovered further utility relating to the method andtools described in connection with FIGS. 1-5. In addition to concernswith stability of a system, such as a power supply, or power converter,a designer is concerned with whether the system will exhibit acceptabletransient characteristics when subjected to a perturbation. By way ofexample, a designer of a power converter apparatus is concerned withtransient voltage and settling time of the system in response toimposition of a step-change of current (i.e., a perturbation of currentapplied to the system). Such concerns are important in order that adesigner may produce a robust and reliable system that can continueoperation within design constraints in the face of line surges or othervariations that may be encountered in a real-life operationalenvironment.

FIG. 6 is an electrical schematic diagram of a power converter apparatusthat is a generalized equivalent to the circuit illustrated in FIG. 1.In FIG. 6, the power converter model proposed by Thottuvelil andVerghese illustrated in FIG. 1 is restated as a power converterapparatus 610 that includes a Thevenin voltage source 612 connected inseries with a characterized impedance 614 and a load impedance 616.Characterized impedance 614 has a value Z_(CL), load impedance 616 has avalue Z_(L). An output voltage V_(OUT) is measured across load impedance616 as indicated in FIG. 6. When characterized impedance 614 is valued.$\begin{matrix}{Z_{CL} = \frac{Z_{OL}}{1 + {AB}}} & \lbrack 6\rbrack\end{matrix}$then power converter apparatus 610 operates substantially the same aspower converter apparatus 10 (FIG. 1).

To evaluate the response of output voltage VOL-T for load impedance 616having various values of Z_(L), Thevenin voltage source 612 is shortedand the equivalent circuit for power converter apparatus 616 isconfigured as illustrated in FIG. 7.

FIG. 7 is an electrical schematic diagram of a power converter of thesort illustrated in FIG. 6, configured for evaluating transient outputvoltage characteristics. In FIG. 7, equivalent power converter apparatus710 includes a characterized impedance 714 and a load impedance 716coupled in parallel intermediate a voltage source (not shown in FIG. 7)connected at a source locus 712 and a ground 718. Characterizedimpedance 714 has a value of Z_(CL); load impedance 716 has a value ofZ_(L). Thus, there is a combined impedance 720 intermediate source locus712 and ground 718 that includes characterized impedance 714 and loadimpedance 716. Combined impedance 720 has a value Z_(COMBINED) that isdefined by: $\begin{matrix}{Z_{COMBINED} = \frac{Z_{CL}{\bullet Z}_{L}}{Z_{CL} + Z_{L}}} & \lbrack 7\rbrack\end{matrix}$

Plotting the various responses of the various impedance values Z_(OL)(open-loop output impedance 15; FIG. 1), closed loop impedance, Z_(L)(load impedance 716; FIG. 7) and Z_(COMBINED) (combined impedance 720;FIG. 7) as a function of frequency yields an interesting revelation.

FIG. 8 is a graphic plot as a function of frequency of variousimpedances in the closed loop system of FIG. 7. In FIG. 8, a graph 800plots impedance values on an impedance axis 802 against frequency valueson a frequency axis 804. A response curve 806 indicates response of openloop impedance values Z_(OL) as a function of frequency. A responsecurve 808 indicates response of closed loop impedance values as afunction of frequency. A response curve 810 indicates response of loadimpedance values Z_(L) as a function of frequency. A response curve 812indicates response of combined impedance values Z_(COMBINED) as afunction of frequency.

Response curve 812 is recognizable by those skilled in the electroniccircuit design arts as being very similar to the response curve for aresonant circuit. Treating response curve as a quasi-resonant circuitand ascribing characteristics of a resonant circuit to response curve812, one observes that response curve 812 exhibits a characteristicfrequency f_(C) and a characteristic impedance R_(C). Characteristicimpedance R_(C) may be represented by an indication of pure resistance,R, because of peculiar properties of resonant circuits when they operateat their resonant frequency.

FIG. 9 is an electrical schematic diagram of a resonant circuit. In FIG.9, a resonant circuit 910 includes a resistive load 912, a capacitiveload 914 and an inductive load 916 coupled in parallel intermediate avoltage source (not shown in FIG. 9) connected at a source locus 918 anda ground 920. Resistive load 912 has a resistive value R, capacitiveload 914 has a capacitive value C and inductive load 916 has aninductive value L. Resonant circuit 910 is sometimes referred to as aRLC circuit referring to the inclusion of a resistive load (R), aninductive load (L) and a capacitive load (C) in the circuit. At theresonant frequency for resonant circuit 910, the phase effects ofcapacitive load 914 and inductive load 916 effectively cancel so thatthe value of resonance for resonant circuit 910 is equal to theresistive load value R. That is, at resonance,[Z]=[R]  [8]Treating response curve 812 as representing a response of a resonantcircuit allows evaluation of response curve 812 using certain attributesof resonant circuits. Resonant circuits are often described in terms ofthe circuit Q, sometimes referred to as the “quality” of the circuit.Circuit Q is a subject that has been the object of much study, andvarious relations among circuit parameters as they relate to Q have beenestablished.

The response of such resonant circuits can be considered to be thesuperposition of two responses: a steady state response and a transientresponse. The transient response describes the behavior of the circuitimmediate following a disturbance and the steady state response is thelong term response after the transient response has subsided.

Application of Ohm's Law to describe the steady state relationship amongcircuit parameters when applying a step-altered current (i.e., a currentthat is substantially instantaneously altered in value) in a circuityields:V _(STEADY STATE) =R _(DC SUPPLY) •I _(STEP)  [9]

FIG. 10 is a graphic plot of signal amplitude as a function of time,illustrating various parameters associated with describing a signalresponse. In FIG. 10, a graphic plot 1000 indicates response of avoltage signal 1014 in terms of voltage amplitude plotted on a voltageaxis 1010 as a function of time, indicated on a time axis 1012. Astep-alteration of current (in this case, a step-increase) is imposed ata time t1, as manifested by an increase of voltage response signal 1014.In particular, voltage response signal 1014 increases from a voltageamplitude V1 to a voltage amplitude value V3. After reaching voltageamplitude value V3, voltage response signal 1014 drops to a lower value,and oscillates for a time before settling at a voltage amplitude valueV2. A first current level supported a voltage response at voltageamplitude V1 until time t1. After imposition of the step-increase incurrent (at time t1) the apparatus for which voltage response signal1014 is relevant experienced a voltage transient displacement having anamplitude (V3−V2), and having a peak value equal to (V3−V1). Thetransient response (i.e., the disturbance in voltage response signal1014) thus recorded lasted for a time following time t1. By a time t2,voltage response signal 1014 had settled (i.e., damped) to a valuewithin a predetermined percentage (for example, 1%-2%) of the steadystate value of voltage response signal 1014 at the higher current level(that is, voltage level V2). The interval (t1-t2) is the settling timefor the circuit experiencing the response indicated by voltage responsecurve 1014. These two parameters—peak transient voltage and settlingtime—are commonly regarded along with circuit stability by systemdesigners in evaluating transient response of a system.

Solving equations that describe RLC circuits to ascertain times ofoccurrences of peak values of R, L and C (e.g., by determining thederivatives of the equations and setting the derivative equations equalto 0) permits determination of time of occurrence of peak voltagevalues. Knowing the time of peak occurrence permits determination of thepeak value of the voltage. By expressing the equations in terms of Qaccording to well-known definition of Q in terms of R, L and C, thefollowing valuable expression is derived from expression [9]:$\begin{matrix}{V_{TRANSIENT} = {I_{STEP}{\bullet R}_{{COMBINED}\quad{CIRCUIT}}\bullet\frac{1}{Q}{\bullet e}^{\lbrack\frac{{- \tan^{- 1}}\sqrt{{4Q} - 1}}{\sqrt{{4Q} - 1}}\rbrack}}} & \lbrack 10\rbrack\end{matrix}$

One expression defining Q that is valuable to the present invention is asolution for Q in terms of phase margin (φ_(m)) provided by Erickson(Fundamentals of Power Electronics; Erickson, Robert; Kluwer AcademicPublishers, Boston Mass.; 1997; p. 336): $\begin{matrix}{Q = \frac{\sqrt{\cos\quad\varphi_{m}}}{\sin\quad\varphi_{m}}} & \lbrack 11\rbrack\end{matrix}$

Substituting expression [11] for Q in expression [10] yields anexpression for V_(TRANSIENT) that is set forth in terms of phase margin(φ_(m)), thereby avoiding employment of Q as a term in the expression.This expression is particularly applicable for use with the presentinvention, as will be described hereinafter at least in connection withFIGS. 13 and 14. Thus, expression [10] may be expressed as:V _(TRANSIENT) =I _(STEP) ·R _(COMBINED CIRCUIT)•Γ(φ_(m))  [12]

-   -   where Γ(φ_(m)) comprises the last term of expression [10] set        forth in terms of phase margin (φ_(m))

The combined resistance of resonant circuit 910 is equal to the combinedimpedance of resonant circuit 910 at characteristic frequency f_(C)(FIG. 8).

It is known, from expression [6] that: $\begin{matrix}{Z_{CL} = \frac{Z_{OL}}{1 + {AB}}} & \lbrack 6\rbrack\end{matrix}$

It, therefore, follows that:Z _(OL) =Z _(CL)(1+AB)  [13]

It is known, from expression [3] that: $\begin{matrix}{{{Loop}{\quad\quad}{Gain}} = {{AB}\frac{Z_{L}}{Z_{L} + Z_{OL}}}} & \lbrack 3\rbrack\end{matrix}$

Substituting Z_(OL) from expression [13] into expression [3] yields:$\begin{matrix}{{{Loop}\quad{Gain}} = {{AB}\frac{Z_{L}}{Z_{L} + {Z_{CL}\left( {1 + {AB}} \right)}}}} & \lbrack 14\rbrack\end{matrix}$

Rearranging expression [14] yields: $\begin{matrix}{Z_{CL} = {Z_{L}\frac{\left\lbrack {\frac{AB}{{Loop}\quad{Gain}} - 1} \right\rbrack}{1 + {AB}}}} & \left\lbrack 15 \right\}\end{matrix}$

It is known, from expression [7] that: $\begin{matrix}{Z_{combined} = \frac{Z_{L}Z_{CL}}{Z_{L} + Z_{CL}}} & \lbrack 7\rbrack\end{matrix}$

Substituting expression [15] into expression [7] yields: $\begin{matrix}{Z_{combined} - \frac{Z_{L}^{2}\left\lbrack {\frac{AB}{{Loop}\quad{Gain}} - 1} \right\rbrack}{Z_{L}\left\lbrack {{AB} + \frac{AB}{{Loop}\quad{Gain}}} \right\rbrack}} & \lbrack 16\rbrack\end{matrix}$

Given that we are interested in the impedance at characteristicfrequency f_(C), that is, whenLoop Gain=e^(Jφ) ^(m)   [17](see expression [4], for a unity gain condition), substitutingexpression [17] into expression [16] and rearranging yields:$\begin{matrix}{Z_{combined} = {Z_{{Lf}{(c)}}\frac{\left\lbrack {1 - \frac{{\mathbb{e}}^{{j\phi}_{m}}}{AB}} \right\rbrack}{1 + e^{{J\phi}_{m}}}}} & \lbrack 18\rbrack\end{matrix}$where Z_(Lf(c)) is the load impedance Z_(L) at the characteristicfrequency, f_(C).

Substituting expression [18] into expression [12] yields:$\begin{matrix}{V_{TRANSIENT} = {I_{STEP}{\bullet Z}_{{Lf}{(c)}}\frac{\left\lbrack {1 - \frac{{\mathbb{e}}^{{j\phi}_{m}}}{AB}} \right\rbrack}{1 + e^{{J\phi}_{m}}}{{\bullet\Gamma}\left( \varphi_{m} \right)}}} & \lbrack 19\rbrack\end{matrix}$

The last two terms of expression [19] are expressed solely in terms ofphase margin (φ_(m)) and characteristic frequency f_(C). For thatreason, expression [19] my be set forth as:V _(TRANSIENT) =I _(STEP) ·Z _(Lf(c)) ·Y(φ_(m) ,f)  [20]

The last term of expression [20] is employed as a multiplier factorexpressed solely in terms of phase margin (φ_(m)) as a function offrequency, f.

FIG. 11 is a graphic plot of multiplier Y as a function of frequency forvarious values of phase margin for a particular power supply. In FIG.11, a graphic plot 1100 indicates response of multiplier Y plotted on amultiplier axis 1110 as a function of frequency plotted on a frequencyaxis 1112, for various phase margin values. Thus, curve 1115 indicatesresponse of multiplier Y for a phase margin of 15 degrees. Curve 1130indicates response of multiplier Y for a phase margin of 30 degrees.Curve 1145 indicates response of multiplier Y for a phase margin of 45degrees. Curve 1160 indicates response of multiplier Y for a phasemargin of 60 degrees. Curve 1175 indicates response of multiplier Y fora phase margin of 75 degrees. Curve 1190 indicates response ofmultiplier Y for a phase margin of 90 degrees.

One may define a settling factor according to the followingrelationship: $\begin{matrix}{{{Settling}\quad{Time}} = \frac{{Settling}\quad{Factor}}{f(c)}} & \lbrack 21\rbrack\end{matrix}$

In such a relationship, the settling factor amounts to an estimate ofthe number of oscillations a signal response takes to damp to apredetermined level.

FIG. 12 is a graphic plot of settling factor at the characteristicfrequency of a system as a function of phase margin. In FIG. 12, agraphic plot 1200 indicates a response curve 1214 for a settling factorplotted on a settling factor axis 1210 and a phase margin axis 1212.

FIGS. 13(A) and (B) illustrate application of one aspect of the methodof the present invention. FIG. 13(A) is a plot of phase responses forvarious phase margins as a function of frequency (lower portion of FIG.13(A)), and impedance magnitude responses for various phase margins as afunction of frequency (upper portion of FIG. 13(A)). FIG. 13(B) is aplot of multiplier Y for various phase margins as a function offrequency. Curve 1115 indicates response of multiplier Y as a functionof frequency at a phase margin of 15 degrees. Curve 1130 indicatesresponse of multiplier Y as a function of frequency at a phase margin of30 degrees. Curve 1145 indicates response of multiplier Y as a functionof frequency at a phase margin of 45 degrees. Curve 1160 indicatesresponse of multiplier Y as a function of frequency at a phase margin of60 degrees. Curve 1175 indicates response of multiplier Y as a functionof frequency at a phase margin of 75 degrees. Curve 1190 indicatesresponse of multiplier Y as a function of frequency at a phase margin of90 degrees. Thus, FIG. 13(B) is a substantially faithful reproduction ofFIG. 11.

FIG. 13(A) is a graphic tool of the sort described in detail inconnection with FIG. 5. A load magnitude response curve 100 is plottedin the upper portion of FIG. 13(A) with phase margin response curves 82a, 84 a, 86 a, 88 a, 90 a, 92 a. A load phase response curve 102 isplotted in the lower portion of FIG. 13(A) with phase margin responsecurves 82 b, 84 b, 86 b, 88 b, 90 b, 92 b. FIG. 13 will be employed toexemplify practice of the method of the present invention.

The method of the present invention preferably begins by noting theintersection of load magnitude response curve 100 with an appropriatephase margin response 82 a, 84 a, 86 a, 88 a, 90 a, 92 a in the upperplot of FIG. 13(A). In exemplary phase margin responses 82 a, 84 a, 86a, 88 a, 90 a, 92 a indicated in FIG. 13, the various phase marginresponses substantially converge at an intersection with load magnituderesponse curve 100. That intersection of load magnitude response curve100 with phase margin responses 82 a, 84 a, 86 a, 88 a, 90 a, 92 aoccurs substantially at a frequency value of 420 Hz, as indicatedgenerally by index 1. Thus, the characteristic frequency Q: of theapparatus being evaluated (the apparatus being evaluated is not shown inFIG. 13) is 420 Hz. If the various phase margin responses 82 a, 84 a, 86a, 88 a, 90 a, 92 a were more divergent in the area of intersection withload magnitude response curve 100, the choice of which phase marginresponse 82 a, 84 a, 86 a, 88 a, 90 a, 92 a to use for determiningcharacteristic frequency f_(C) requires determining the frequency valuethat intersects load magnitude response curve 100 and a given phasemargin response curve (or an interpolated value for phase marginresponse) in the upper plot of FIG. 13(A) and also intersects load phaseresponse curve 102 and the same phase margin response curve (orinterpolated value for phase margin) in the lower plot of FIG. 13(A).

Having ascertained characteristic frequency f_(C) using the upper plotof FIG. 13(A), the method next determines the point at whichcharacteristic frequency F_(C) intersects load phase response curve 102in the lower plot of FIG. 13(A). In the exemplary plot of FIG. 13,characteristic frequency f_(C) 420 Hz intersects load phase responsecurve 102 at a phase margin value of 38 degrees, as generally indicatedby index 2. Continuing with the exemplary practice of the method of thepresent invention illustrated in FIG. 13, the next step involves usingcharacteristic frequency f_(C) and the phase margin. In order to do so,one returns to the upper plot of FIG. 13(A) to ascertain intersection ofcharacteristic frequency f_(C) (f_(C)=420 Hz) with the phase marginresponse locus (φ_(m)=38 degrees) to ascertain load impedance atcharacteristic frequency Z_(Lf(c)). The upper plot of FIG. 13 (A) yieldsa value for impedance at characteristic frequency Z_(Lf(c) of) 0.037ohms, as generally indicated by index 3.

The next step in the exemplary practice of the method of the presentinvention illustrated in FIG. 13 requires addressing FIG. 13(B) usingcharacteristic frequency F_(C) (f_(C) 420 Hz) to ascertain intersectionof characteristic frequency f_(C) with the phase margin value (φ_(m)=38degrees) obtained as described, and generally indicated by index 3.

Referring to FIG. 13(B), one may observe that multiplier Y has a valueindicated by a characteristic frequency F_(C) of 420 Hz with a phasemargin φ_(m) of 38 degrees. The value so indicated for multiplier Y is0.65, as generally indicated by index 4.

Substituting values for characteristic frequency f_(C), phase marginφ_(m), load impedance at characteristic frequency Z_(Lf(c)) andmultiplier Y into expression [20] yields:V _(TRANSIENT) =I _(STEP) •Z _(Lf(c)) •Y(φ_(m)).  [20] V _(TRANSIENT) =I _(STEP)•(0.037)•(0.65)  [22]V _(TRANSIENT) =I _(STEP)•(0.24)  [23]

Thus, peak transient voltage V_(TRANSIENT) may be straightforwardlydetermined in terms of step change of current.

FIGS. 14(A) and (B) illustrate application of a second aspect of themethod of the present invention. In FIG. 14(A), a response curve 1406 isplotted on a voltage amplitude axis 1402 as a function of time indicatedon a time axis 1404. Response curve 1406 illustrates a transient voltageresponse that undergoes a perturbation at a time t1. Response curve 1406settles to within a defined acceptable percentage (not specified in FIG.14(A)) of a steady state level at a time t2. The time interval (t1−t2)is the settling time for the apparatus for which response curve 1406 isrepresentative.

In FIG. 14(B), settling factor Y is plotted in a response curve 1414against a settling factor axis 1410 as a function of phase margin(φ_(m)) indicated on a phase margin axis 1412. Continuing the exemplaryperformance of the method of the present invention begun in connectionwith FIG. 13, the method next requires entering FIG. 14(B) at a valuefor phase margin at characteristic frequency PM_(C) to determine thepoint at which phase margin at characteristic frequency PM_(C)intersects response curve 1414. The intersection point indicatessettling factor at characteristic frequency SF_(C). In the exemplaryillustration of FIG. 14(B), a phase margin at characteristic frequencyPM_(C) equal to 38 degrees (as previously determined in connection withFIG. 13) intersects response curve 1414 at a point corresponding to avalue for settling factor at characteristic frequency SF_(C) equal to1.9, as generally indicated by index 5. Substituting values for settlingfactor at characteristic frequency SF_(C) and characteristic frequencyF_(C) into expression [21] yields: $\begin{matrix}{{{Settling}\quad{Time}} = \frac{SettlingFactor}{f(c)}} & \lbrack 21\rbrack \\{{{Settling}\quad{Time}} = {\frac{1.9}{420\quad{Hz}} = {4.5\quad{milliseconds}}}} & \lbrack 24\rbrack\end{matrix}$

Response curves of the sort illustrated in FIGS. 5, 13 and 14 areembodiments of reference tools that can easily be created for a closedloop apparatus or product. The graphic reference tools in FIGS. 5, 13and 14 are very useful in evaluating closed loop apparatuses forstability and transient response (e.g., peak transient voltage andsettling time). It should be kept in mind that such graphicmanifestations of the reference tools of the present invention areillustrative, and are not intended to limit embodying apparatuses of thepresent invention in a graphic form. Reference tools according to thepresent invention may advantageously be embodied in on-line forms thatpermit rapid, precise and repeated evaluations using computer programssubstantially emulating the method of the present invention.

The method and apparatus of the present invention are important designtools that can be used to determine whether or not a particular powersupply or other closed loop apparatus is appropriate for an application,or whether a given load will be stable and exhibit appropriate transientresponse characteristics with a particular power supply. In the past,such evaluative determinations have required extensive systemevaluations to iteratively determine whether one particular apparatus oranother apparatus would more suitably accommodate a particular load.Such evaluations were often conducted by the apparatus manufacturerbased upon load characterizations provided by the customer. Significantdelays in development were experienced by customers in awaitingevaluation results from the manufacturer.

The load-independent characteristics of the method and apparatus of thepresent invention are of particular value because the method andapparatus of the present invention facilitate evaluation of apparatusesvis-à-vis particular loads by the customer without any need to awaitevaluations and testing by the manufacturer.

It is to be understood that, while the detailed drawings and specificexamples given describe preferred embodiments of the invention, they arefor the purpose of illustration only, that the apparatus and method ofthe invention are not limited to the precise details and conditionsdisclosed and that various changes may be made therein without departingfrom the spirit of the invention which is defined by the followingclaims.

1. A method for predicting at least one transient responsecharacteristic of a closed loop apparatus; said closed loop apparatushaving an open loop impedance, a design load impedance, an outputvoltage and at least one inherent internal gain; the method comprisingthe steps of: (a) identifying an impedance scaling factor associatedwith said closed loop apparatus; said impedance scaling factor beingexpressed in terms including said open loop impedance, said at least oneinherent internal gain, a gain variable and a phase variable; (b)vectorally establishing a first scaling value for said impedance scalingfactor as a function of frequency while maintaining a first variable ofsaid gain variable and said phase variable at a first working value torecord said first scaling value for a plurality of frequencies; (c)vectorally establishing a second scaling value for said impedancescaling factor as a function of frequency while maintaining a secondvariable of said gain variable and said phase variable at a secondworking value to record said second scaling value for a plurality offrequencies; (d) creating a first reference tool relating said firstscaling value with said second variable of said gain variable and saidphase variable as a function of frequency; (e) creating a secondreference tool relating said second scaling value with said firstvariable of said gain variable and said phase variable as a function offrequency; (f) identifying a combined impedance for said apparatus; saidcombined impedance involving said closed loop impedance, said at leastone internal gain and said design load impedance; (g) determining acombined impedance response of said combined impedance as a function offrequency; (h) determining a characteristic frequency for said combinedimpedance response; said characteristic frequency occurringsubstantially at a peak value of said combined impedance response; (i)employing at least one of said first reference tool and said secondreference tool to establish a first design value for one parameter ofphase margin and said design load impedance for said closed loopapparatus at said characteristic frequency; (j) employing at least oneof said first reference tool and said second reference tool to establisha second design value for the other parameter of phase margin and saiddesign load impedance than said one parameter for said closed loopapparatus at said characteristic frequency; (k) vectorally establishinga transient multiplier factor for said closed loop apparatus; saidtransient multiplier factor being associated with said output voltage ofsaid closed loop apparatus with said design load impedance as a functionof frequency for selected values of phase margin; (l) creating a thirdreference tool relating said transient multiplier factor with saidselected values of phase margin as a function of frequency; (m)employing said third reference tool to establish a third design valuefor said transient multiplier factor associated with said characteristicfrequency and at least one of said first design value and said seconddesign value; and (n) mathematically combining at least two of saidfirst design value, said second design value and said third design valuewith a design step current according to a predetermined relation toestablish transient voltage excursion of said output voltage in responseto an applied step current substantially at said characteristicfrequency.
 2. A method for predicting at least one transient responsecharacteristic of a closed loop apparatus as recited in claim 1 whereinthe method comprises the further steps of: (o) treating said combinedimpedance response as relating to a resonant circuit to establishing aquasi-Q factor for said closed loop apparatus; (p) establishing asettling factor for said closed loop apparatus; said settling factorbeing associated with said quasi-Q factor; (q) creating a fourthreference tool relating said settling factor with phase margin for saidclosed loop apparatus; (r) employing said fourth reference tool toestablish a design settling factor associated with at least one of saidfirst design value and said second design value; and (s) mathematicallycombining said design settling factor with said characteristic frequencyaccording to a predetermined relation to establish settling time of saidclosed loop apparatus in response to said applied step current.
 3. Amethod for predicting at least one transient response characteristic ofa closed loop apparatus as recited in claim 1 wherein at least one ofsaid first reference tool, said second reference tool and said thirdreference tool is created in a multidimensional graphic form.
 4. Amethod for predicting at least one transient response characteristic ofa closed loop apparatus as recited in claim 1 wherein at least one ofsaid first reference tool, said second reference tool and said thirdreference tool is created as a multidimensional vectoral table.
 5. Amethod for predicting at least one transient response characteristic ofa closed loop apparatus as recited in claim 4 wherein saidmultidimensional vectoral table is maintained on-line.
 6. A method forpredicting at least one transient response characteristic of a closedloop apparatus as recited in claim 2 wherein at least one of said firstreference tool, said second reference tool, said third reference tooland said fourth reference tool is created in a multidimensional graphicform.
 7. A method for predicting at least one transient responsecharacteristic of a closed loop apparatus as recited in claim 2 whereinat least one of said first reference tool, said second reference tool,said third reference tool and said fourth reference tool is created as amultidimensional vectoral table.
 8. A method for predicting at least onetransient response characteristic of a closed loop apparatus as recitedin claim 7 wherein said multidimensional vectoral table is maintainedon-line.
 9. A method for predicting at least one transient responsecharacteristic of a closed loop apparatus; said closed loop apparatushaving an open loop impedance, a design load impedance, an outputvoltage and at least one inherent internal gain; the method comprisingthe steps of: (a) providing a first reference tool associated with saidapparatus; said first reference tool relating impedance response of saidapparatus, independent of said design load impedance, with a firstvariable of a gain variable and a phase variable; (b) providing a secondreference tool associated with said apparatus; said second referencetool relating impedance response of said apparatus, independent of saiddesign load impedance, with a second variable of a gain variable and aphase variable other than said first variable; (c) determining acombined impedance response for said apparatus as a function offrequency; said combined impedance response involving said open loopimpedance, said design load impedance and said at least one inherentinternal gain; (d) employing at least one of said first reference tooland said second reference tool to establish a first design value for oneparameter of said phase variable and said design load impedance for saidclosed loop apparatus at a characteristic frequency; said characteristicfrequency occurring substantially at a peak value of said combinedimpedance response; (e) employing at least one of said first referencetool and said second reference tool to establish a second design valuefor the other parameter of said phase variable and said design loadimpedance than said one parameter for said closed loop apparatus at saidcharacteristic frequency; (f) vectorally establishing a transientmultiplier factor for said closed loop apparatus; said transientmultiplier factor being associated with said output voltage of saidclosed loop apparatus with said design load impedance as a function offrequency for selected values of phase margin; (g) creating a thirdreference tool relating said transient multiplier factor with saidselected values of phase margin as a function of frequency; (h)employing said third reference tool to establish a third design valuefor said transient multiplier factor associated with said characteristicfrequency and at least one of said first design value and said seconddesign value; and (i) mathematically combining at least two of saidfirst design value, said second design value and said third design valuewith a design step current according to a predetermined relation toestablish transient voltage excursion of said output voltagesubstantially at said characteristic frequency in response to applyingsaid design step current.
 10. A method for predicting at least onetransient response characteristic of a closed loop apparatus as recitedin claim 9 wherein the method comprises the further steps of: (i)treating said combined impedance response as relating to a resonantcircuit to establishing a quasi-Q factor for said closed loop apparatus;(k) establishing a settling factor for said closed loop apparatus; saidsettling factor being associated with said quasi-Q factor, (l) creatinga fourth reference tool relating said settling factor with phase marginfor said closed loop apparatus; (m) employing said fourth reference toolto establish a design settling factor associated with at least one ofsaid first design value and said second design value; and (n)mathematically combining said design settling factor with saidcharacteristic frequency according to a predetermined relation toestablish settling time of said closed loop apparatus in response tosaid applied step current.
 11. A method for predicting at least onetransient response characteristic of a closed loop apparatus as recitedin claim 9 wherein at least one of said first reference tool, saidsecond reference tool and said third reference tool is created in amultidimensional graphic form.
 12. A method for predicting at least onetransient response characteristic of a closed loop apparatus as recitedin claim 9 wherein at least one of said first reference tool, saidsecond reference tool and said third reference tool is created as amultidimensional vectoral table.
 13. A method for predicting at leastone transient response characteristic of a closed loop apparatus asrecited in claim 12 wherein said multidimensional vectoral table ismaintained on-line.
 14. A method for predicting at least one transientresponse characteristic of a closed loop apparatus as recited in claim10 wherein at least one of said first reference tool, said secondreference tool, said third reference tool and said fourth reference toolis created in a multidimensional graphic form.
 15. A method forpredicting at least one transient response characteristic of a closedloop apparatus as recited in claim 10 wherein at least one of said firstreference tool, said second reference tool, said third reference tooland said fourth reference tool is created as a multidimensional vectoraltable.
 16. A method for predicting at least one transient responsecharacteristic of a closed loop apparatus as recited in claim 15 whereinsaid multidimensional vectoral table is maintained on-line.
 17. Anapparatus for predicting at least one transient response characteristicof a closed loop device; said closed loop device having an open loopimpedance, a design load impedance, an output voltage and at least oneinherent internal gain; the apparatus comprising: (a) a first referencetool relating a first scaling value of an impedance scaling valueassociated with said closed loop device with a first design variable ofa gain variable and a phase variable as a function of frequency relatingto said closed loop device, said impedance scaling factor beingexpressed in terms independent of said load; said first scaling valuebeing vectorally established as a function of frequency whilemaintaining other design variables than said first design variable at atleast one working value to record said first scaling value for aplurality of frequencies; (b) a second reference tool coupled with saidfirst reference tool; said second reference tool relating a secondscaling value of said impedance scaling factor associated with saidclosed loop device with a second design variable of a gain variable anda phase variable as a function of frequency; said second scaling valuebeing vectorally established as a function of frequency whilemaintaining said first design variable at a second working value torecord said second scaling value for a plurality of frequencies; (c) athird reference tool coupled with at least one of said first referencetool and said second reference tool; said third reference tool relatinga transient multiplier factor with selected values of phase margin ofsaid closed loop device as a function of frequency; said transientmultiplier factor being associated with said output voltage of saidclosed loop device with said design load impedance as a function offrequency for said selected values of phase margin.
 18. An apparatus forpredicting at least one transient response characteristic of a closedloop device as recited in claim 17 wherein the apparatus furthercomprises: (d) a fourth reference tool coupled with at least one of saidfirst reference tool, said second reference tool and said thirdreference tool; said fourth reference tool relating a settling factorwith phase margin for said closed loop device; said settling factorbeing associated with a quasi-Q factor established for a combinedimpedance response of said closed loop device; said combined loopresponse involving said closed loop impedance, said at least oneinternal gain and said design load impedance; said combined impedanceresponse being treated as a resonant circuit response when determined asa function of frequency for establishing said quasi-Q factor.
 19. Anapparatus for predicting at least one transient response characteristicof a closed loop device as recited in claim 17 wherein at least one ofsaid first reference tool, said second reference tool and said thirdreference tool is created in a multidimensional graphic form.
 20. Anapparatus for predicting at least one transient response characteristicof a closed loop device as recited in claim 17 wherein at least one ofsaid first reference tool, said second reference tool and said thirdreference tool is created as a multidimensional vectoral table.
 21. Anapparatus for predicting at least one transient response characteristicof a closed loop device as recited in claim 20 wherein saidmultidimensional vectoral table is maintained on-line.
 22. An apparatusfor predicting at least one transient response characteristic of aclosed loop device as recited in claim 18 wherein at least one of saidfirst reference tool, said second reference tool, said third referencetool and said fourth reference tool is created in a multidimensionalgraphic form.
 23. An apparatus for predicting at least one transientresponse characteristic of a closed loop device as recited in claim 18wherein at least one of said first reference tool, said second referencetool, said third reference tool and said fourth reference tool iscreated as a multidimensional vectoral table.
 24. An apparatus forpredicting at least one transient response characteristic of a closedloop device as recited in claim 23 wherein said multidimensionalvectoral table is maintained on-line.